Remarks on solitary waves in equations with nonlocal cubic terms
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914843924103168 |
|---|---|
| author | Marstrander, Johanna Ulvedal |
| author_facet | Marstrander, Johanna Ulvedal |
| contents | In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \begin{equation*}
\partial_t u + \partial_x(Λ^s u + uΛ^r u^2) = 0, \end{equation*} where $Λ^s, Λ^r$ are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, $s>0$, while the operator on the nonlinear part is assumed to act slightly smoother, $r<s-1$. The problem is related to the mathematical theory of water waves; we build upon previous works on similar equations, extending them to allow for a nonlocal nonlinearity. Mathematical tools include constrained minimization, Lion's concentration-compactness principle, spectral estimates, and product estimates in fractional Sobolev spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15148 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Remarks on solitary waves in equations with nonlocal cubic terms Marstrander, Johanna Ulvedal Analysis of PDEs 76B15, 76B25, 35A15 In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \begin{equation*} \partial_t u + \partial_x(Λ^s u + uΛ^r u^2) = 0, \end{equation*} where $Λ^s, Λ^r$ are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, $s>0$, while the operator on the nonlinear part is assumed to act slightly smoother, $r<s-1$. The problem is related to the mathematical theory of water waves; we build upon previous works on similar equations, extending them to allow for a nonlocal nonlinearity. Mathematical tools include constrained minimization, Lion's concentration-compactness principle, spectral estimates, and product estimates in fractional Sobolev spaces. |
| title | Remarks on solitary waves in equations with nonlocal cubic terms |
| topic | Analysis of PDEs 76B15, 76B25, 35A15 |
| url | https://arxiv.org/abs/2406.15148 |